square (prime+square)

Dean Hickerson dean at math.ucdavis.edu
Fri Mar 28 23:05:15 CET 2003

Ralf Stephan (ralf at ark.in-berlin.de) asked:

> the oeis, with my help, came up with the following conjecture. 
> Is it trivial?
> %N A075555 Smallest prime p such that p+n is a square, 0 if no such p seems
>            to exist, 1 if it has been proven.
> %C A075555 Conjecture: a(n)=0 iff n=A047845(i)^2, i.e. there are no squares
>            of the form p+k^2, p prime, 2k+1 composite.

The "i.e." seems to be unjustified.  As others have noted, it's trivial that
if n is the square of an element of A047845 then a(n)=0.  But it's not clear
that those are the only values of n for which a(n)=0.  It would follow from
Schinzel's hypothesis, but I don't believe this special case has been proved.

Dean Hickerson
dean at math.ucdavis.edu

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